We consider a geometrically nonlinear model for crystal plasticity in two dimensions, with two active slip systems and rigid elasticity. We prove that the rank-1 convex envelope of the condensed energy density is obtained by infinite-order laminates, and express it explicitly via the 2F1 hypergeometric function. We also determine the polyconvex envelope, leading to upper and lower bounds on the quasiconvex envelope. The two bounds differ by less than 2%.

The first part of this paper is devoted to a study of the classical bifurcation problem in a Hilbert space, under the assumption that the operators involved are gradient operators, but not necessarily compact. Our approach to the problem was introduced by Krasnosel'skii, but here we show that his assumption about the compactness of the operators can be replaced by a much weaker Lipschitz type condition, without affecting the generality of his conclusions.

The rest of the paper is concerned with the analogous problem when the operator is knownto be asymptotically linear rather than Fréchet differentiable. Indeed, we show that this question can always be reduced to the first case, after some manipulation. After this manipulation the new operator is found to be a Fréchet differentiable gradient operator, and so we can invoke the results of the first part. This manipulation is in the spirit of that of [11] but is necessarily different.

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. We solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular, we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore, we construct examples in which the genericity assumption does not hold and indicate that the situation is different for these.

This paper is concerned with the effect of the delays on the bounded solutions of the nth order n ≧ 1) differential equation

Sufficient conditions involving the retardations τJ(j = 1,2, …, m) which insure that every bounded solution of the considered equation is oscillatory are given. The results obtained generalise recent ones in [3 and 4].

We give a new and elementary proof of the known result: a non-constant mapping of finite distortion f : Ω ⊂ ℝn → ℝn is discrete and open, provided that its distortion function if n = 2 and that for some p > n − 1 if n ≥ 3.

The entire collection comprises from 150 to 200 species, the majority of which are new to science.

As yet only some of the more abnormal Synascidiœ and about half the Ascidiœ simplices have been carefully examined. The present paper is the preliminary report on the ASCIDIÆ, the first family of the Ascidiœ simplices.

We show that the space of Sturm–Liouville operators characterized by H = (q, α, β) ∈ L1 (0, 1) × [0, π)2 such that is homeomorphic to the partition set of the space of all admissible sequences which form sequences that converge to q, α, and β individually. This space, Γ, of quasi-nodal sequences is a superset of, and is more natural than, the space of asymptotically nodal sequences defined in Law and Tsay (On the well-posedness of the inverse nodal problem. Inv. Probl.17 (2001), 1493–1512). The definition of Γ relies on the L1 convergence of the reconstruction formula for q by the exactly nodal sequence.

We derive an asymptotic expansion for the Titchmarsh–Weyl m-function associated with the second order linear differential equation

in the case where the only restriction on the real-valued function q is

In the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

In many problems in physics and chemistry certain determinants of large order and of a special type require to be evaluated. The determinants in question usually arise from some kind of secular equation, and in most cases they owe their origin to a system of particles of one kind or another which, in their equilibrium positions, form a regular lattice, each particle being acted upon by its nearest neighbours and perhaps by a fixed boundary. These forces may be assumed to be elastic in character, to the degree of approximation required. For simplicity we shall refer to these particles as atoms although they may be electrons, molecules or even material particles in the Newtonian sense, according to the nature of the problem under investigation. In particular we may mention the occurrence of these determinants in problems involving the solution of Schrödinger's Wave Equation for permitted energy levels and in determining the distribution of electric charge in crystals, metals and large molecules. They have also been used by Born in his investigations of crystal structure by means of X-rays. It is hoped therefore that the results achieved in this paper will be of interest to the physicist and chemist as well as to the mathematician.

Travelling wavefronts for a system of two reaction–diffusion equations are studied. The existence of a family of wavefronts (one for each wave speed) as well as the existence of the minimal speed (in the case that the Lewis number is greater than $1$) are proved. Asymptotic formulae for the wavefronts are established. New results are obtained when applying the main theorems to an isothermal autocatalytic chemical reaction system.

At times when processes of great ontogenetic importance are taking place, the interaction of heredity and environment presents problems of considerable interest. Laying as it does the foundations of future independent existence, prenatal development must be the concern of all those interested in the mental and physical suitability of human beings for the environment in which they have to live. Owing to lack of information, pregnancy from the point of view of the fœtus has perforce been largely neglected by the physician and sociologist, but the need to remedy this defect in our knowledge had been clearly indicated by Hogben (1931). As an of example of a field of research that would repay much attention, the work of Pearson (1914) and S. Hansen (1920) may be mentioned. These author wrote suggestively of the handicap of the first-born, using extensive data on the incidence of disease, mental defect, and the physical qualities of the new-born child in relation to birth order.

The action of the symmetric group $S_4$ on the tetrahedron algebra, introduced by Hartwig and Terwilliger, is studied. This action gives a grading of the algebra which is related to its decomposition into a direct sum of three subalgebras isomorphic to the Onsager algebra. The ideals of both the tetrahedron algebra and the Onsager algebra are determined.

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium. The estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighbouring the stable relative equilibrium in question and with any prefixed (spatio-temporal) isotropy subgroup. Moreover, it is easily computable in particular examples. It is interesting to see how, in our result, the existence of non-trivial relative periodic orbits requires (generic) conditions on the higher-order terms of the Taylor expansion of the Hamiltonian function, in contrast with the purely quadratic requirements of the Weinstein–Moser theorem, which emphasizes the highly nonlinear character of the relatively periodic dynamical objects.

In this paper we show that each quasiperiodic standing wave solution of the real Ginzburg–Landau equation which is on the global branch emanating from the Eckhaus unstable periodic orbit is itself unstable. A rigorous proof of the instability is given by showing that the linearised operator about such a solution has spectrum which contains an interval along the unstable axis of the spectral plane. The proof employs some geometric and topological methods arising from a dynamical systems approach to the analysis of the eigenvalue problem for the linearised operator.

The paper considers the small time evolution of the interface which appears in the boundary value problem

A diffusion dominated small time outer expansion is not uniformly valid when x = O(t½ log t) and has to be supplemented by an inner expansion valid in a region where diffusion and absorption are of equal importance. For p < 1 a zero order inner solution is constructed, which is consistent with a moving interface where u is zero, and the small time development of this is given.

These results are first derived in a heuristic way using the method of matched expansions. However, an important aim of the paper is to give rigorous proofs, being one of the very few cases in nonlinear diffusion where this has been done.

This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.

It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.

We establish upper and lower bounds for various norms of solutions and their gradients for the equation ut = div (|∇u|m−1 ∇u) in ℝN in terms of the norms of the initial data. Based on the L∞ estimate of ∇u, we conclude that u(x, t) is Lipschitz continuous in space-time, for all t>0, whenever u(x,0) is in L1(ℝN).

We consider the non-linear problem −Δu(x)−f(x, u(x)) = λu(x) for x ∈ℝN and u ∈ W1,2(ℝN). We show that, under suitable conditions on f, there exist infinitely many branches all bifurcating from the lowest point of the continuous spectrum λ = 0. The method used in the proof is based on a theorem of Ljusternik-Schnirelman type for the free case.

The switched diffusion process associated with a weakly coupled system of elliptic equations is studied via a Dirichlet space approach and is applied to prove the existence theorem of the Cauchy initial problem for the system. A representation theorem for the solution of the Dirichlet boundary value problem and a generalised Skorohod decomposition for the reflecting switched diffusion process are obtained.